Theorem dsndxnbasendx

Description: The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024)

	Ref	Expression
Assertion	dsndxnbasendx	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne {\mathrm{Base}}_{\mathrm{ndx}}$$

Proof

Step	Hyp	Ref	Expression
1		1re	$$⊢1\in ℝ$$
2		1nn	$$⊢1\in \mathbb{N}$$
3		2nn0	$$⊢2\in {\mathbb{N}}_0$$
4		1nn0	$$⊢1\in {\mathbb{N}}_0$$
5		1lt10	$$⊢1<10$$
6	2 3 4 5	declti	$$⊢1<12$$
7	1 6	gtneii	$$⊢12\ne 1$$
8		dsndx	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)=12$$
9		basendx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}=1$$
10	8 9	neeq12i	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne {\mathrm{Base}}_{\mathrm{ndx}}\leftrightarrow 12\ne 1$$
11	7 10	mpbir	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne {\mathrm{Base}}_{\mathrm{ndx}}$$